Coordination by Option Contracts in a Retailer-Led Supply Chain with

574
quantities ( q1, q 2)
that maximize his own expected
profit, subject to the retailer’s ordering behavior. His
decision on the second-stage production is relatively
straightforward, i.e.,
* * *
q2 = max{ d − q1,0}
.
The difficulty lies in his decision on q 1 . Obviously
he should at least produce d 1 , but should he produce
more? The answer depends on his estimate on the likelihood
of the retailer’s second order. For such a decision
his problem is
ε ( d1)
max Π ( q ) = wd − cq + v( q − d )d G(
ε)
+
q1d1 S 1 1 1 1 1
0
1 1
ε ( q1
)
∫ ε ( d1)
∞
wd 2 2 vq1 d G
∫
ε ( q1
)
∫
[ + ( − )]d ( ε ) +
[ wd −c( d−q)]d G(
ε ).
2 2 2 1
The solution to this problem is given by Lemma 3.
Lemma 3 Given the retailer’s ordering behavior,
the supplier will maximize his expected profit by
setting
* *
q1 d1 q′ 1
* * *
= max{ , } and q2 = max{ d − q1,0},
where 1 q′ is implied by c2 − c1
G( ε ( q′
1))
= .
c2−v Because this problem has a newsvendor structure,
the first order condition works. The proof is straightforward
so the details are omitted. These problems
have reviewed the benchmark and decentralized uncoordinated
system performance. Although the precise
relationships between * *
d 1 , q 1,
and c
q 1 depending on
factors like the ratio of the two production costs, the
ratio of the two wholesale prices, and the estimate of
the market signal cannot be determined, the channel is
certainly not coordinated. This is implied by Lemma 2
which states that for a given market signal the retailer’s
total order quantity is less than the optimal total inventory
required in a centralized system. The next section
focuses on how an option mechanism coordinates
the supply chain and realizes profit allocation.
2 Channel Coordination
To coordinate the channel, the retailer must choose the
proper contract parameters so that the supplier’s production
quantities in each period are equal to the opti-
* c
mal solutions in the centralized system, i.e., q = q
and
1 1
* c
q = q . To do so the retailer must give incentives
Tsinghua Science and Technology, August 2008, 13(4): 570-580
to the supplier to overproduce in each period. Recall
that q1d1 and q2 max{ d − q1,0}
. Thus d 1 and
max{ d − q1,0}
are the minimum production number
for the supplier’s production in each period. The coordinating
contract is defined as follows: The option contract
has three parameters of two option prices o 1 and
o 2 and one exercise price e . The option prices are
the unit compensation for the supplier’s production
quantities beyond the minimums in each period. When
the realized demand exceeds the retailer’s total order,
he will purchase the supplier’s excess inventory, if any,
to meet the market demand at a unit price e .
The sequence of events is summarized as follows:
(1) At the beginning of period 1, the retailer sets a
firm order d 1 . The retailer also announces an option
contract with option prices o 1 and o 2 and exercise
price e. The supplier, in turn, decides to produce
q1 d1
of goods in the cheap mode which will produce
an income of o1( q1− d1)
.
(2) At the beginning of period 2, after the market
signal is observed, the retailer orders d 2 . The supplier
selects a second-period production, q2 max{ d− q1,0}
,
that involves the expensive mode and produces an income
of o2( q2 −max{ d − q1,0})
.
(3) At the beginning of the selling season, after the
demand is realized, the retailer purchases additionally
m= min{max{ D−d,0}, q− d}
number of goods at
unit price e . The supplier delivers all that the retailer
wants. The problem assumes that the supplier is obligated
to fill the retailer’s entire order and can neither
sell directly to the final customer (i.e., the retailer’s
customer) nor supply inventory in excess of the retailer’s
total order quantity.
With the option contract, the supplier’s problem
becomes
max Π ( q ) = −( c − o ) q + ( w − o ) d +
q1d1 S 1 1 1 1 1 1 1
∫
0
∞
Π ( ε, q , q )d G(
ε),
S 1 2
where ΠS( ε, q1, q2) = max πS( ε,
q1, q2)
and
q2max{ d−q1,0} π ( ε , q , q ) = − c q + o ( q −max{ d − q ,0}) +
S 1 2 2 2 2 2 1
d
wd + vq ( + q − d)d F( x|
ε ) +
∫
2 2
0
1 2
q1+ q2
d
∫
[( ex− d) + vq ( − x)]d Fx ( | ε ) +